L(s) = 1 | + (0.492 − 0.159i)2-s + (1.13 − 1.56i)3-s + (−1.40 + 1.01i)4-s + (−0.809 − 2.08i)5-s + (0.309 − 0.951i)6-s + (−1.96 − 2.70i)7-s + (−1.13 + 1.56i)8-s + (−0.226 − 0.696i)9-s + (−0.732 − 0.896i)10-s + 3.34i·12-s + (−4.03 + 1.31i)13-s + (−1.40 − 1.01i)14-s + (−4.17 − 1.10i)15-s + (0.761 − 2.34i)16-s + (−3.67 − 1.19i)17-s + (−0.222 − 0.306i)18-s + ⋯ |
L(s) = 1 | + (0.348 − 0.113i)2-s + (0.655 − 0.902i)3-s + (−0.700 + 0.509i)4-s + (−0.362 − 0.932i)5-s + (0.126 − 0.388i)6-s + (−0.743 − 1.02i)7-s + (−0.401 + 0.552i)8-s + (−0.0754 − 0.232i)9-s + (−0.231 − 0.283i)10-s + 0.965i·12-s + (−1.11 + 0.363i)13-s + (−0.374 − 0.272i)14-s + (−1.07 − 0.284i)15-s + (0.190 − 0.585i)16-s + (−0.891 − 0.289i)17-s + (−0.0524 − 0.0722i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0451877 - 0.868643i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0451877 - 0.868643i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 + 2.08i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.492 + 0.159i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.13 + 1.56i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (1.96 + 2.70i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (4.03 - 1.31i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.67 + 1.19i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.39 - 2.46i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 3.48iT - 23T^{2} \) |
| 29 | \( 1 + (5.60 - 4.07i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.69 + 8.30i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.05 + 1.45i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.40 - 1.01i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 6.45iT - 43T^{2} \) |
| 47 | \( 1 + (-6.73 + 9.26i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.06 + 0.671i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (1.02 - 0.745i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.62 + 11.1i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 2.20iT - 67T^{2} \) |
| 71 | \( 1 + (2.53 - 7.79i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.87 - 3.96i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.452 - 1.39i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (9.41 + 3.05i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 0.464T + 89T^{2} \) |
| 97 | \( 1 + (-8.69 + 2.82i)T + (78.4 - 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.981881457655591945259120290680, −9.186417217081785304591533101688, −8.441506968961362914384029749015, −7.46876718002587691754621934994, −7.10087773866498220095340926611, −5.41749555235088971031321467376, −4.38406808546355396873246281164, −3.59362395513781100965056573886, −2.21747578086721487156285065916, −0.38700845213472870861955269231,
2.66257392408947118968091361996, 3.43776713313567852399389379721, 4.45219507830560331212672895858, 5.48498020609963718015959693774, 6.43708850854929244199290642360, 7.51560016619151278825934490563, 8.852192441256593690761023763327, 9.380958524691371058306165131769, 9.968263199955718679921424420296, 10.83638654700060095245015523030